# Properties

 Label 2736.2.k.h Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{5} -2 \beta q^{7} +O(q^{10})$$ $$q + 2 q^{5} -2 \beta q^{7} -\beta q^{11} + 2 \beta q^{13} + 2 q^{17} + ( -1 - 3 \beta ) q^{19} -\beta q^{23} - q^{25} -5 \beta q^{29} + 6 q^{31} -4 \beta q^{35} + 8 \beta q^{37} -3 \beta q^{41} -5 \beta q^{47} - q^{49} -7 \beta q^{53} -2 \beta q^{55} + 8 q^{59} -8 q^{61} + 4 \beta q^{65} -2 q^{67} + 8 q^{71} -4 q^{77} -8 q^{79} -9 \beta q^{83} + 4 q^{85} + 7 \beta q^{89} + 8 q^{91} + ( -2 - 6 \beta ) q^{95} -2 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} + 4q^{17} - 2q^{19} - 2q^{25} + 12q^{31} - 2q^{49} + 16q^{59} - 16q^{61} - 4q^{67} + 16q^{71} - 8q^{77} - 16q^{79} + 8q^{85} + 16q^{91} - 4q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 1.41421i − 1.41421i
0 0 0 2.00000 0 2.82843i 0 0 0
2431.2 0 0 0 2.00000 0 2.82843i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.h 2
3.b odd 2 1 912.2.k.a 2
4.b odd 2 1 2736.2.k.i 2
12.b even 2 1 912.2.k.d yes 2
19.b odd 2 1 2736.2.k.i 2
24.f even 2 1 3648.2.k.c 2
24.h odd 2 1 3648.2.k.f 2
57.d even 2 1 912.2.k.d yes 2
76.d even 2 1 inner 2736.2.k.h 2
228.b odd 2 1 912.2.k.a 2
456.l odd 2 1 3648.2.k.f 2
456.p even 2 1 3648.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.a 2 3.b odd 2 1
912.2.k.a 2 228.b odd 2 1
912.2.k.d yes 2 12.b even 2 1
912.2.k.d yes 2 57.d even 2 1
2736.2.k.h 2 1.a even 1 1 trivial
2736.2.k.h 2 76.d even 2 1 inner
2736.2.k.i 2 4.b odd 2 1
2736.2.k.i 2 19.b odd 2 1
3648.2.k.c 2 24.f even 2 1
3648.2.k.c 2 456.p even 2 1
3648.2.k.f 2 24.h odd 2 1
3648.2.k.f 2 456.l odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} - 2$$ $$T_{7}^{2} + 8$$ $$T_{11}^{2} + 2$$ $$T_{31} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$8 + T^{2}$$
$11$ $$2 + T^{2}$$
$13$ $$8 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$19 + 2 T + T^{2}$$
$23$ $$2 + T^{2}$$
$29$ $$50 + T^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$128 + T^{2}$$
$41$ $$18 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$50 + T^{2}$$
$53$ $$98 + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$( 2 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$162 + T^{2}$$
$89$ $$98 + T^{2}$$
$97$ $$8 + T^{2}$$